The instantaneous compactification phenomenon is the one where a solution \(u(x,t)\), \(x\in \mathbb R^N\), \(0\leq t\leq T\), of a Cauchy problem has a compact support for any \(t>0\), though the initial function is nonzero on the whole space \(\mathbb R^N\). The author finds conditions for the instantaneous compactification for the equation \[ \frac{\partial}{\partial t}\left(| u| ^{\beta -1}u\right)-\sum\limits_{i=1}^N \frac{\partial}{\partial x_i}\left(| \frac{\partial u}{\partial x_i}| ^{p_i-2} \frac{\partial u}{\partial x_i}\right)+| u| ^{\lambda -1}u=0 \] where \(\beta >0\), \(p_i>1+\beta\) (\(i=1,\dots ,N\)), \(0<\lambda <\beta\). Earlier this problem was studied for the isotropic case (\(p_1=p_2=\cdots =p_N\)); see, for example, S. P. Degtyarev [Sb. Math. 199, No. 4, 511–538 (2008); translation from Mat. Sb. 199, No. 4, 37–64 (2008; Zbl 1172.35036)].